Analytic first order nonadiabatic coupling matrix elements of spin-adapted open-shell time-dependent density functional theory

This paper presents the derivation, implementation, and benchmarking of analytic first-order nonadiabatic coupling matrix elements for the spin-adapted X-TDDFT method, demonstrating that it significantly reduces errors compared to standard U-TDDFT and provides qualitatively correct insights into the photophysics of open-shell systems like copper(II) porphyrin.

The Gist

Imagine you are trying to predict how a complex machine, like a spinning top made of magnets, will react when you shake it. In the world of chemistry, this "machine" is a molecule with unpaired electrons (like organic radicals or transition metal complexes), and the "shake" is light hitting it, causing it to jump into an excited state.

Scientists use a tool called Time-Dependent Density Functional Theory (TDDFT) to simulate these reactions. Think of TDDFT as a sophisticated weather forecast for molecules. It predicts how the molecule moves and changes energy.

However, there's a problem. Standard TDDFT (let's call it U-TDDFT) is like a weather forecast that assumes the wind always blows in a straight line. It works okay for simple molecules, but for complex ones with "unpaired" electrons (like our spinning magnet top), it gets confused. It treats the two "spins" of the electrons (let's call them Spin A and Spin B) as if they are independent, which leads to errors. It's like trying to describe a dance where two partners are holding hands, but the forecast assumes they are dancing alone.

The New Solution: X-TDDFT

The authors of this paper developed an upgrade called X-TDDFT. This is like a new weather model that understands the partners are holding hands. It forces the math to respect the "spin" rules of quantum mechanics. They had already used this to predict the energy and shape of these molecules better, but they were missing one crucial piece: Nonadiabatic Coupling Matrix Elements (NACMEs).

What is a NACME?
Imagine the molecule is a car driving on a bumpy road.

• Energy tells you how fast the car is going.
• Gradients tell you which way the road is sloping.
• NACMEs tell you how likely the car is to jump lanes or crash into a different state.

In chemistry, this "lane jumping" is called Internal Conversion (IC). It's the process where a molecule absorbs energy, gets excited, and then quickly dumps that energy back down to the ground state, often releasing heat instead of light. If your NACME calculation is wrong, you might think the car will stay in its lane, when in reality, it's about to swerve violently into a ditch.

What Did They Do?

The team derived the mathematical formulas to calculate these "lane-jumping" probabilities (NACMEs) using their new, spin-aware X-TDDFT method. They then tested it in two ways:

1. The Small Test (Formaldehyde Radical): They compared their new method against a "gold standard" super-accurate calculation (like checking a new GPS against a satellite map). They found that the old method (U-TDDFT) was often wrong by a huge margin—sometimes off by one-third to two-thirds. The new method (X-TDDFT) fixed most of these errors, making the prediction of how fast the molecule "cools down" (Internal Conversion rate) much more accurate. In some cases, the new method predicted the cooling speed was 100 times slower than the old method predicted.

2. The Big Test (Copper Porphyrins): They looked at complex copper-based molecules (similar to the heme in blood, but with copper).

   • The Old View (U-TDDFT): Predicted that when the molecule gets excited, it has an equal chance of cooling down directly or taking a detour through intermediate states.
   • The New View (X-TDDFT): Predicted that the molecule almost never cools down directly. It almost always takes the detour.
   • The Result: This completely changed the story of how these molecules behave. The old method didn't just get the numbers slightly wrong; it got the story wrong. It also messed up the comparison between different versions of the molecule (with different chemical decorations), making it look like one version was faster than another when the opposite was true.

The Takeaway

The paper concludes that for molecules with unpaired electrons (like radicals or transition metals), you cannot trust the old "straight-line" math (U-TDDFT) for predicting how they switch energy states.

Just as you wouldn't use a flat map to navigate a mountain range, you shouldn't use the old TDDFT method for these complex molecules. The new X-TDDFT method acts like a 3D topographical map, revealing that the "roads" (energy pathways) are very different than previously thought. This is crucial for scientists trying to design better solar cells, LEDs, or catalysts, because if you don't know which "lane" the molecule will jump into, you can't control its behavior.

In short: The authors built a better ruler to measure how molecules "jump" between energy states. They proved that the old ruler was so inaccurate that it was telling completely different stories about how these molecules work, especially for those involving copper and other transition metals.

Technical Summary

Technical Summary: Analytic First-Order Nonadiabatic Coupling Matrix Elements of Spin-Adapted Open-Shell Time-Dependent Density Functional Theory

Problem Statement
Spin-adapted time-dependent density functional theory (X-TDDFT) has recently been established as a robust method for calculating excitation energies and analytic gradients of open-shell molecules (such as radicals and transition metal complexes), offering significant improvements over unrestricted TDDFT (U-TDDFT) by properly handling spin contamination and implicitly including specific double excitations. However, the effect of spin-adaptation on nonadiabatic coupling matrix elements (NACMEs)—critical for simulating internal conversion (IC) rates and nonadiabatic dynamics—remained unknown. While rigorous U-TDDFT NACME theories exist, they suffer from large errors due to spin contamination, particularly for states dominated by closed-shell-to-vacant-shell (CV) excitations. Existing spin-adapted NACME theories are either limited to semiempirical Hamiltonians, specific spin-flip excitations, or lack the necessary formalism for general open-shell systems. Consequently, accurate simulation of photophysics and photochemistry in open-shell systems using spin-conserving frameworks was hindered by the lack of a consistent X-TDDFT NACME theory.

Methodology
The authors derive and implement analytic first-order NACMEs for the X-TDDFT method, specifically for ground-state-to-excited-state (ge) and excited-state-to-excited-state (ee) transitions. The derivation follows an equation-of-motion (EOM) approach, replacing the standard random phase approximation (RPA) excitation and de-excitation operators with their spin-adapted counterparts.

Key theoretical components include:

1. Spin-Adapted Operators: The formalism utilizes a restricted open-shell Kohn-Sham (ROKS) reference state. The excitation operator $O^\dagger_I$ is constructed using spin-adapted CV(0) and CV(1) excitations, where the latter is mixed with specific double excitations ($\Psi^{\bar{t}a}_{\bar{i}t}$) to ensure proper spin symmetry.
2. Lagrangian Formulation: To handle the implicit dependence of excitation vectors and molecular orbital (MO) coefficients on nuclear coordinates, a Lagrangian approach is employed. This allows for the calculation of analytic derivatives without explicitly differentiating the MO coefficients, incorporating Pulay terms via Lagrange multipliers ($Z$ and $W$ vectors).
3. Correction Terms: The X-TDDFT NACMEs are expressed as the standard U-TDDFT NACMEs plus specific correction terms arising from:
   • The use of a spin-adapted reference (ROKS) instead of an unrestricted one (UKS).
   • The modification of the CV(1) excitation operator to include double excitations.
   • The resulting changes in the transition density matrices (TDMs) and the Casida matrix ($A$ and $B$ blocks).
4. Implementation Strategy: The implementation is designed to be built upon existing U-TDDFT NACME codes. The authors demonstrate that the additional computational overhead is acceptable, comparable to that of analytic gradient calculations, as it primarily involves solving modified Z-vector equations and calculating intermediate density matrices.

Key Contributions

• Theoretical Derivation: The paper provides the first rigorous derivation of analytic NACMEs for spin-conserving X-TDDFT, covering both ge and ee transitions.
• Algorithmic Implementation: The method is implemented in the BDF software package. It leverages existing U-TDDFT infrastructure, requiring only the addition of specific correction terms to the transition density matrices and the Lagrangian multipliers.
• Benchmarking: The authors benchmark the method against high-level multireference calculations (XMS-CASPT2) for the formaldehyde radical cation ($CH_2O^+$).
• Application to Transition Metal Complexes: The method is applied to copper(II) porphyrins (CuP, CuOEP, CuTPP) to simulate excited-state relaxation pathways and internal conversion rates.

Results

• Benchmark Accuracy ($CH_2O^+$): Compared to XMS-CASPT2 reference data, X-TDDFT NACMEs reduce the mean absolute deviation (MAD) of U-TDDFT NACME norms by approximately 32% to 65% (depending on the functional). The most significant improvements are observed for states with significant CV(1) character, where U-TDDFT often fails due to spin contamination. However, the authors note that for certain mixed states (e.g., $1^2A_2$), X-TDDFT can occasionally yield larger errors than U-TDDFT, though this is attributed to fortuitous error cancellation in the U-TDDFT results rather than a failure of the spin-adapted theory.
• Copper(II) Porphyrins:
  • NACME Magnitudes: X-TDDFT predicts significantly smaller NACME magnitudes (up to an order of magnitude) for the $2T_1 \to 2S_0$ transition compared to U-TDDFT. This reduction is attributed to the near-perfect cancellation of $\alpha$ and $\beta$ contributions in pure CV(1) states within the spin-adapted framework, a cancellation that is incomplete in U-TDDFT due to orbital shape differences.
  • Internal Conversion Rates: The reduction in NACME norms leads to drastic reductions in calculated IC rates (up to two orders of magnitude).
  • Relaxation Pathways: For unsubstituted CuP, U-TDDFT predicts that the dominant relaxation pathway involves roughly equal direct IC to the ground state and indirect IC via ligand-field states. In contrast, X-TDDFT predicts that the indirect pathway via ligand-field states is dominant.
  • Substituent Effects: The relative trends of IC rates for substituted porphyrins (CuP, CuOEP, CuTPP) differ between the two methods. U-TDDFT predicts a trend of CuOEP < CuTPP < CuP, whereas X-TDDFT predicts CuTPP < CuOEP < CuP. This indicates that U-TDDFT errors are not only large but also unsystematic, failing to capture relative trends even for closely related molecules.

Significance and Claims
The paper claims that the development of X-TDDFT NACMEs completes the essential toolkit for studying photophysical and photochemical processes of open-shell molecules using spin-adapted TDDFT. The authors emphasize that spin-adaptation is not merely a correction for energies and gradients but is crucial for NACMEs and, consequently, for the accuracy of simulated nonadiabatic dynamics and IC rates.

The study concludes that relying on U-TDDFT NACMEs for open-shell systems can lead to qualitatively incorrect predictions of relaxation mechanisms and quantitative errors in rate constants that are both large and unsystematic. Therefore, the authors recommend the use of X-TDDFT NACMEs for open-shell systems generally, even in cases where U-TDDFT and X-TDDFT excitation energies appear similar. The work lays the foundation for future applications in nonadiabatic molecular dynamics and the study of ultrafast photophysics in radicals and transition metal complexes.